Question: Giselle works as a carpenter and as a blacksmith. She earns $\$20$ per hour as a carpenter and $\$25$ per hour as a blacksmith. Last week, Giselle worked both jobs for a total of $30$ hours, and earned a total of $\$690$. How long did Giselle work as a carpenter last week, and how long did she work as a blacksmith? Giselle worked as a carpenter for
Solution: Let $x$ represent the time Giselle worked as a carpenter and let $y$ represent the time she worked as a blacksmith. Since we have two unknowns, we need two equations to find them. Let's use the given information in order to write two equations containing $x$ and $y$. For instance, we are given that Giselle earns $\$\textit{20}$ per hour as a carpenter, $\$\textit{25}$ per hour as a blacksmith, and earned a total of $\$\textit{690}$ last week. How can we model this sentence algebraically? The total amount of money Giselle made as a carpenter can be modeled by $20x$, and the total amount of money she made as a blacksmith can be modeled by $25y$. Since these together add up to $\$690$, we get the following equation: $20x+25y=690$ We are also given that last week, Giselle worked as a carpenter and a blacksmith for a total of $\textit{30}$ hours. This can be expressed as: $ x + y =30$ Let's rewrite this equation so that it's solved for $x$ : $x = 30-y$ Now that we have a system of two equations, we can go ahead and solve it! Let's substitute $ x={30- y}$ into the first equation: $\begin{aligned}20 x+25 y &= 690\\\\ 20 \cdot ({30-y})+25y&=690\\\\ 600-20y+25y&=690\\\\ 5y &=90\\\\ y&=18\end{aligned}$ Now we can substitute $y = 18$ into $x=30- y$ and find that $x=12$. Recall that $x$ denotes the time Giselle worked as a carpenter and $y$ denotes the time she worked as a blacksmith. Therefore, Giselle worked as a carpenter for $\textit{12}$ hours and as a blacksmith for $\textit{18}$ hours last week.